IEV: Calculating Expected Offspring
For a random variable taking integer values between 1 and , the expected value of is . The expected value offers us a way of taking the long-term average of a random variable over a large number of trials.
As a motivating example, let be the number on a six-sided die. Over a large number of rolls, we should expect to obtain an average of 3.5 on the die (even though it’s not possible to roll a 3.5). The formula for expected value confirms that .
More generally, a random variable for which every one of a number of equally spaced outcomes has the same probability is called a uniform random variable (in the die example, this “equal spacing” is equal to 1). We can generalize our die example to find that if is a uniform random variable with minimum possible value and maximum possible value , then . You may also wish to verify that for the dice example, if is the random variable associated with the outcome of a second die roll, then .
Given: Six nonnegative integers, each of which does not exceed 20,000. The integers correspond to the number of couples in a population possessing each genotype pairing for a given factor. In order, the six given integers represent the number of couples having the following genotypes:
Return: The expected number of offspring displaying the dominant phenotype in the next generation, under the assumption that every couple has exactly two offspring.
1 0 0 1 0 1
library(dplyr) f <- "iev.txt" pop <- data_frame( p = c(1, 1, 1, 0.75, 0.5, 0), n = readLines(f) %>% strsplit(split = " ") %>% unlist() %>% as.numeric() ) sum(2 * pop$n * pop$p) %>% cat()